38
I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points.
A triangle has many ways you can think about it. ...
24
One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...
21
Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes are basic to our understanding of everything.
Where mathematics comes from, by George Lakoff and Rafael E. Núñez. Basic Books, 2000.
Exploring the role of ...
21
I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points.
If $P = (-1,-1)$, $Q =...
19
I don't see this as a major issue, nor do I believe that the word "unique" is in any particular need of saving. There are a large number of terms in mathematics which correspond to vernacular English words, but which have distinct technical meaning in mathematics. For example, an "odd" number is an integer which is not a multiple of $2$,...
17
Perhaps not pointing out that the obvious steps are obvious but that the insights are insights.
I believe students don't feel bad for not seeing the "magic steps" by themselves, so pointing out that those are hard is not a problem. The opposite is what you mention: they would feel bad for not seeing the obvious. Hence, only treat the difficult as difficult.
16
Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such:
It appears that even many serious professional mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive ...
13
One of the most colorful names I have heard is the Chicken Mc Nugget theorem:
for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$.
link1, link2.
From the links:
The story goes that the Chicken McNugget Theorem got its name because in McDonalds, ...
13
So German "$b$ kürzt sich weg" becomes in English "$b$ cancels out". We may also say "$b$ is eliminated".
12
One thing that you might want to do early on in your course is think about the classroom norms that you wish to establish. From your post, it seems like an example of a norm in your class is that it is okay to ask a question of the form:
What do you mean by obvious (routine, etc)?
If you do not try to make it clear to students from the outset that such ...
11
In Central Mexico, the expression
\begin{equation}
x_{\pm} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}
that solves quadratic equations of the form $ax^2 + bx + c = 0$ is called "fórmula del chicharronero" (formula of the chicharronero).
The chicharronero is the guy who sells salty snacks made of wheat (called chicharrones). Outside most schools ...
11
My phrase has always been math "statement". Equations and inequalities clearly assert/state a relationship between two or more things.
My go-to direction using this would be something like: "Clearly explain the meaning of each math statement below in the context of the problem you just read." Then I list things like $f(4)=3$ or $g(7)>g(...
11
The ambiguous answer is relatively correct (actually you need to know how old they are at the start of the problem). But each fission is an individual splitting, not a generation.
Perhaps this:
A certain species of bacteria splits into two cells ( a process called fission) every 20 minutes (the "generation" length), assuming proper growth ...
10
For presenting proofs, I would only use terminology such as "obviously", "clearly", "routine", etc if it would be for a course just below the class had the student had 'perfect' recollection. For instance, in a introductory commutative algebra class, if the proof should be routine for the previous regular algebra class (i.e., something involving a course ...
10
How about the shoelace formula for the area of an arbitrary
simple polygon?
(Image from Wikipedia.)
The formula computes the area from the coordinates of the vertices,
essentially by a cross product to compute (signed) areas of triangles.
10
If you continue the operations until
$$\require{cancel}\frac{x}{b}=\frac{c}{b},\qquad\left(\frac{x}{b}\right)b=\left(\frac{c}{b}\right)b,\qquad x\left(\frac{b}{b}\right)=c\left(\frac{b}{b}\right),\qquad x\left(\frac{\cancel b}{\cancel b}\right)=c\left(\frac{\cancel b}{\cancel b}\right)$$
then $x=c$, I would say that you cancelled the common factor $b$ in the ...
8
If you simplify a term by adding and subtracting something you call this a "nahrhafte Null" in German (probably translates to "nutritious null"?).
8
I often refer to the identities $(AB)^{-1} = B^{-1}A^{-1}$ or $(AB)^T = B^TA^T$ as the socks-shoes identity. I'm not sure how wide-spread this is, I certainly did not invent it and I'm pretty sure I've read at least one of these in at least one text.
8
In Russian, the Squeeze Theorem (a.k.a. The Pinching Theorem) is called "Теорема о двух милиционерах" — "Two Policemen Theorem". The idea is that if two policemen are holding a criminal between them, the bad guy is going to the same place, probably jail or precinct, where the policemen are going.
8
In general, as others have noted, if you have an equation such as
$$\frac{x}{b}=\frac{c}b$$
The step to get from there to
$$x=c$$
is typically referred to as cancelling the denominator. More generally, if you can just remove some piece of the equation, you can use the verb "cancel" both with that piece as the object and the subject. For instance:
We cancel ...
8
I confess, when you start talking about 1-D triangles, my own first thought is "how can you have non-colinear points in 1-D?". So, I imagine most students that age will have a far more difficult time with that.
Keep in mind age appropriateness. For 9-12 year old children, you are generally looking at a level of psychological development ...
8
Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the interior of a plane region was relevant to problem solving at that grade level. For these types of elementary shapes, the boundary and the interior completely ...
7
Some examples:
"One plus five squared" could be read as $(1+5)^2$ or $1+5^2$. This is a well-known ambiguity in natural language, for example in the following sentence
I saw a man on a hill with a telescope.
we don't know if it was a man who had the telescope, or the speaker, or perhaps the telescope was just standing on some hill.
See also this list.
...
7
The argument has been made that this is sort of a misappropriation of the terms, because the levels are meant to define levels of understanding rather than levels of detail. I'm assuming what you're aiming for is that you can say to your student(s) "I'm looking for a level 2 explanation here", and they would provide you with an answer that would ...
7
To my mind the defintion "Every x has a unique f(x)" of one-to-one is problematic because "has a unique" is neither clear English nor precise. The definition is usually stated as "$f(x) = f(y)$ implies $x = y$", which is unambiguous and avoids any potential interpretative problems related to the use of the word "unique. Alternatively, one could define $f$ to ...
7
Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution.
I agree with you here, and I think this is the key point. To me they are clearly well-defined: "Triangle", "square", and polygons in general, are bounded regions on the Euclidean plane, i.e., 2D figures.
...
6
My elementary students always wanted to know the name of the symbol shown here:
We called it the division house as did many of my colleagues, but my students wanted a mathematical name. We therefore wrote to Dr. Math at Drexel. We were told there is no name and were referred to this paper.
I subsequently held a contest (on election day) and the winner ...
6
iff and if
In my experience, students who have a solid grasp of first-order logic have absolutely no problem with the inconsistent use of "if" in definitions. The problem is that most students don't, and as I've personally observed, often confuse between "if" and "iff" precisely because of such notational issues. Therefore the ...
6
I am comfortable saying "Solve the equation $x+2$=4" and also saying "Using the equation $(a+b)(a-b)=a^2-b^2$, we see that...".
On other other hand I would only ever speak of solving an equation, not an identity, and I would be willing to use the word "identity" in my second example above.
So I think I would say that an identity is a kind of equation, one ...
6
I've just remembered that "Donkey Theorem" is used to refer to triangle inequality in geometry textbooks in Iran. The name implies that even a donkey which is on one corner of a triangle chooses the straight path (rather than the broken one) to get to the other corner where there is some hay to eat.
I checked to see if it is used elsewhere and I learned ...
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